Question: For a given positive integer $n > 2^3$, what is the greatest common divisor of $n^3 + 3^2$ and $n + 2$?
Notice that by the sum of cubes factorization, $n^3 + 8 = (n+2)(n^2 - 2n + 4)$ is an integer divisible by $n+2$. Thus,
\begin{align*}
\text{gcd}\,(n^3 + 9, n+2) &= \text{gcd}\,(n^3 + 9 - (n^3 + 8), n+2) \\ 
& = \text{gcd}\,(1,n+2) \\
& = \boxed{1}.
\end{align*}